As an instructor, how to use the first day of class?
Students basically want to know if they should take your class. To that end I would include
- A short (10-15 minute) sales pitch explaining what exactly your class is about and why your topic is interesting.
- Administrative details of the class (I would cover this after the sales pitch, so students who are late don't miss anything).
- A presentation on the first topic in your syllabus. This is important because hearing your first lecture will give them a good idea of the difficulty of the class, which prerequisites are required, and the style/quality of your lecturing. Also, if your entire first lecture is a sales pitch, students will feel like they wasted time coming to your class, or that your class is "easy"/not serious.
My perspective is as a student of physics; I hope this answer generalizes sufficiently.
I have had several teachers in physics who have presented on the first day whichever tricky mathematical theorem, process, or derivation will be used most frequently throughout the semester. This becomes valuable because each time the concept comes up during the rest of the course, the teacher can say, "Now I'm using the Helmholtz theorem (or whatever), which you'll recall from our first class," and the distinction of having been the very first material presented means the students actually do remember it, or at least remember of it. Whereas a proof done sometime in the middle of the fourth lecture will as likely as not need to be reviewed each time it recurs, because the students don't recognize it when it resurfaces.
So if the item is well-chosen, it can save enormous amounts of time in the presentation of later material, since it can essentially be skipped each time it comes up with a casual "…as you'll remember from our first class" in a way that material introduced later often sadly cannot.
Obviously you have to determine if there is a similarly valuable sort of thing in whatever field you teach.
From a relatively recent student point of view, once you have discussed the syllabus, university policies, and your rules for e.g. how homework should be turned in, try to make as much as possible of the class typical of what the rest of the course is going to be like. Think of it as a sample.
Students may be faced with having to make a decision early in the session on which classes to take. The more typical the first class is, the more valid data they have to make that decision. If, on the other hand, you spend the whole of the first class doing things other than teaching in your normal style for the class, they have no way to evaluate whether that style works for them.